The six-time grand slam winner even had an animated discussion with a coach Juan Carlos Ferrero after losing the second set.
Norrie took full advantage, sealing a 4-6 6-3 6-4 victory on his second match point with a strong first serve that Alcaraz returned long. It was his first career win against a No.1-ranked player.
Alcaraz made 54 unforced errors and won only 64 per cent of his first-serve points as he slipped to a third loss in eight matches against the British player. It was their first meeting indoors.
The defeat on Tuesday (local time) also ended Alcaraz's 17-match winning run in Masters events and means second-ranked Jannik Sinner will move to the top of the ATP rankings if he wins the tournament. Sinner plays Zizou Bergs on Wednesday.
Norrie next faces the winner of Wednesday's match between cousins Valentin Vacherot and Arthur Rinderknech.
Vacherot beat 14th-seeded Jiri Lehecka 6-1 6-3 in the first round earlier on Tuesday. The 40th-ranked Vacherot faces Rinderknech a little more than two weeks after beating him in the Shanghai Masters final.
Vacherot made an against-the-odds run from qualifying in China to win his first career tournament.
Preparations will be a little different this time.
"We'll stay in our own bubbles. We're cousins, yes, but we're not on the same team so we're not going to have dinner together tonight," said Vacherot, who is from Monaco.
"In Shanghai we had breakfast together, we warmed up together, because we were far from everyone and everywhere.
"He came to see my matches, I would come and watch his matches, and we had no one else to rely upon. (This time) all our families are here, so we'll stick to our sides, and I hope that we're going to deliver a wonderful match."
Fifth-seeded American Ben Shelton reached the third round with a 7-6 (4) 6-3 win over Italy's Flavio Cobolli. He next faces Russian Andrey Rublev.
In the late match, fourth-seeded American Taylor Fritz was to take on Australia's Aleksandar Vukic, who battled his way through qualifying to make the main draw.
In other first-round play, there were wins for ninth-seeded Felix Auger-Aliassime and No. 11 Daniil Medvedev.
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